Subluminal and superluminal velocities of free-space photons

Konstantin Y. Bliokh

arxiv: 2602.17576 · v3 · submitted 2026-02-19 · 🪐 quant-ph · physics.optics

Subluminal and superluminal velocities of free-space photons

Konstantin Y. Bliokh This is my paper

Pith reviewed 2026-05-15 20:41 UTC · model grok-4.3

classification 🪐 quant-ph physics.optics

keywords wavepacketsgroup velocityphase velocitysubluminalsuperluminalfree-space propagationelectromagnetic beamsphoton velocities

0 comments

The pith

Spatially localized wavepackets in free space always travel with a group velocity below c and a phase velocity above c whose product equals c squared.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that any spatially localized electromagnetic wavepacket propagating in a straight line through empty space carries a group velocity slower than light and a phase velocity faster than light. Their product remains fixed at c squared regardless of the packet's shape or frequency content. This duality is presented as an intrinsic feature of localization itself rather than an effect of any surrounding medium. The result is derived through classical field theory, scalar wave equations, and a quantum-mechanical photon description, with explicit checks on Gaussian beams and higher-order modes. A reader would care because the finding separates the energy transport speed from the phase oscillation speed without invoking special relativity violations.

Core claim

Rectilinear free-space propagation of spatially localized electromagnetic wavepackets is characterized by a subluminal group velocity and a superluminal phase velocity whose product equals c squared. These velocities coincide with the energy velocity and momentum velocity, respectively. The relation holds across electromagnetic field theory, scalar wavepacket evolution, and quantum photon wavefunction treatments, and is verified by direct calculation for Gaussian and higher-order beams.

What carries the argument

The fixed product of group velocity and phase velocity equaling c squared for any spatially localized wavepacket.

If this is right

Energy transport through a localized packet always occurs at the subluminal group velocity.
The superluminal phase velocity corresponds to the momentum velocity and does not carry information.
The same velocity product appears in both classical and quantum descriptions of the photon.
Higher-order beams and packets obey the identical relation as fundamental Gaussian modes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result may clarify why some pulsed-beam experiments appear to show superluminal features while preserving causality.
Analogous velocity duality could be tested in acoustic or matter-wave packets under similar localization conditions.
Photon arrival-time statistics in quantum optics might need reinterpretation once group and phase contributions are separated this way.

Load-bearing premise

The wavepackets remain perfectly rectilinear and spatially localized without dispersion or boundary interactions that would change the velocity product.

What would settle it

A precision measurement of both group and phase velocities for a freely propagating, spatially localized Gaussian wavepacket in vacuum that finds their product differing from c squared.

read the original abstract

We consider rectilinear free-space propagation of electromagnetic wavepackets using electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals $c^2$. These velocities are also known as the 'energy' and 'momentum' velocities, introduced by K. Milton and J. Schwinger. We illustrate general conclusions by explicit calculations for Gaussian and higher-order beams and wavepackets, and also highlight subtleties of the quantum-mechanical description based on the 'photon wavefunction'.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows localized free-space wavepackets have subluminal group velocity and superluminal phase velocity with product exactly c² via standard plane-wave sums, with concrete Gaussian examples, but the invariance under diffraction needs checking.

read the letter

The core result is that rectilinear free-space wavepackets carry a group velocity below c and phase velocity above c whose product is c², presented as energy and momentum velocities from Milton and Schwinger. The paper supplies explicit calculations for Gaussian and higher-order beams plus notes on the photon wavefunction, which makes the standard superposition argument concrete rather than purely formal. Those beam examples are the useful part: they let a reader plug in numbers and see the velocities emerge directly from the transverse k components. The quantum section flags a few subtleties without overclaiming, which is appropriate for the subfield. The stress-test concern about diffraction is worth taking seriously. The derivations appear anchored at a reference plane where the transverse spectrum is fixed; free-space propagation spreads the beam and shifts the effective k_perp distribution, so the extracted velocities become z-dependent. The manuscript does not appear to propagate the packets far from the waist or quantify how much the product deviates outside the Rayleigh range. If the full text contains such checks they would close the gap; otherwise the claim reads as local rather than strictly inherent for the entire packet. This is solid enough for optics and quantum photonics readers who already work with localized beams and want the velocity definitions spelled out with examples. It is not a broad advance but the calculations are reproducible from the given setup. I would send it to peer review so the diffraction point can be clarified in revision; the math is standard and the illustrations are explicit, so referee time is justified.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that spatially localized electromagnetic wavepackets undergoing rectilinear free-space propagation are inherently characterized by a subluminal group velocity and a superluminal phase velocity whose product equals c². These are identified with the energy and momentum velocities of Milton and Schwinger. The claim is supported via electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism, with explicit calculations for Gaussian and higher-order beams and wavepackets, plus discussion of photon-wavefunction subtleties.

Significance. If the result is shown to hold invariantly under propagation, the work would clarify the interpretation of phase and group velocities for localized free-space packets and link them directly to conserved energy-momentum quantities. The provision of explicit calculations for standard beam profiles and the use of multiple independent formalisms constitute a concrete strength.

major comments (1)

[Explicit calculations for Gaussian and higher-order beams] The explicit calculations for Gaussian and higher-order beams (and the supporting derivations) are performed at a fixed reference plane, typically the waist. Because free-space diffraction evolves the transverse spectrum, increasing the effective k_⊥ distribution and rendering β(z) position-dependent, the manuscript must propagate the packets and verify that the product v_g v_p remains exactly c² at arbitrary z. Without this demonstration the 'inherently characterized' claim for rectilinear propagation is not yet established.

minor comments (1)

[Quantum-mechanical formalism] The quantum-mechanical section would benefit from a brief explicit statement of how the phase and group velocities are extracted from the photon wavefunction to make the correspondence with the classical results fully transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the multi-formalism approach and the explicit calculations. We address the major comment below and will revise the manuscript to incorporate the requested demonstration.

read point-by-point responses

Referee: [Explicit calculations for Gaussian and higher-order beams] The explicit calculations for Gaussian and higher-order beams (and the supporting derivations) are performed at a fixed reference plane, typically the waist. Because free-space diffraction evolves the transverse spectrum, increasing the effective k_⊥ distribution and rendering β(z) position-dependent, the manuscript must propagate the packets and verify that the product v_g v_p remains exactly c² at arbitrary z. Without this demonstration the 'inherently characterized' claim for rectilinear propagation is not yet established.

Authors: We agree that verifying the relation at arbitrary propagation distances is necessary to fully substantiate the claim of inherent characterization during rectilinear free-space propagation. Although the general electromagnetic field theory, scalar wavepacket analysis, and quantum-mechanical formalism are derived without reference to a specific plane and rely on the fixed transverse wavevector spectrum together with the dispersion relation that enforces v_g v_p = c² independently of z, the explicit illustrations were indeed given at the waist for simplicity. We will revise the manuscript by propagating the Gaussian and higher-order wavepackets to selected planes z ≠ 0, recomputing the local group and phase velocities, and confirming that the product remains exactly c². These additional calculations will be added to the relevant sections. revision: yes

Circularity Check

0 steps flagged

No circularity: velocities follow algebraically from free-space dispersion on superposed plane waves

full rationale

The central claim follows directly from the linear dispersion relation ω = c |k| applied to any superposition with nonzero transverse wave-vector components k_⊥. For each spectral component the longitudinal wave number is k_z = sqrt((ω/c)^2 − k_⊥²), so the phase velocity ω/k_z and group velocity c² k_z/ω are related by the algebraic identity v_g v_p = c². This identity is an immediate consequence of the dispersion relation itself and does not rely on any fitted parameter, self-referential definition, or prior result by the same authors. Explicit Gaussian and higher-order examples are simply evaluations of the same identity at a fixed reference plane; they are not predictions that are forced by construction. No load-bearing step reduces to a self-citation or to an ansatz smuggled in from earlier work. The derivation is therefore self-contained against external benchmarks (Maxwell equations in free space) and receives the default non-circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of standard electromagnetic field theory, scalar wave equations, and the applicability of a photon wavefunction in free space; no new free parameters, ad-hoc constants, or invented entities are introduced.

axioms (2)

standard math Standard electromagnetic field theory and Maxwell equations govern free-space propagation.
Basis for the rectilinear propagation analysis.
domain assumption A quantum-mechanical photon wavefunction description is valid for the wavepackets considered.
Invoked with noted subtleties in the abstract.

pith-pipeline@v0.9.0 · 5393 in / 1341 out tokens · 28377 ms · 2026-05-15T20:41:06.009362+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals c²
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

using the paraxial approximation for the longitudinal wavevector component, kz ≃ k − k⊥²/2k

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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